Modern Control Engineering

Section 7–8 / Closed-Loop Frequency Response of Unity-Feedback Systems 479

• 4 – 3 – 2 – 1 01 2X

M= 1.2 Y

M= 1.3 M=^1

M= 1.4

M= 1.6

M= 2.0

M= 3.0

M= 5.0

• 1


• 2

1

2

M= 0.8

M= 0.4

M= 0.6

Figure 7–81
A family of constant
Mcircles.

If Equation (7–22) can be written

If the term M^2 /AM^2 -1B

2

is added to both sides of this last equation, we obtain

(7–23)

Equation (7–23) is the equation of a circle with center at X=–M^2 /AM^2 -1B,Y=0

and with radius @M/AM^2 -1B@.

The constant Mloci on the G(s)plane are thus a family of circles. The center and ra-

dius of the circle for a given value of Mcan be easily calculated. For example, for

M=1.3, the center is at (–2.45, 0)and the radius is 1.88. A family of constant Mcir-

cles is shown in Figure 7–81. It is seen that as Mbecomes larger compared with 1, the

Mcircles become smaller and converge to the –1+j0point. For M>1, the centers of

theMcircles lie to the left of the –1+j0point. Similarly, as Mbecomes smaller com-

pared with 1, the Mcircle becomes smaller and converges to the origin. For 0<M<1,

the centers of the Mcircles lie to the right of the origin.M=1corresponds to the locus

of points equidistant from the origin and from the –1+j0point. As stated earlier, it is

a straight line passing through the point and parallel to the imaginary axis. (The

constantMcircles corresponding to M>1lie to the left of the M=1line, and those

corresponding to 0<M<1lie to the right of the M=1line.) The Mcircles are sym-

metrical with respect to the straight line corresponding to M=1and with respect to the

real axis.

A-^12 ,0B

aX+

M^2

M^2 - 1

b

2

+Y^2 =

M^2

AM^2 - 1 B^2

X^2 +

2M^2

M^2 - 1

X+

M^2

M^2 - 1

+Y^2 = 0

MZ1,